Please explain Bell's theorem

[septimus]

@glowacks

I’m not seeing where GHZ leads to anything particularly more definite than the standard Bell’s inequality tests. It looks like it derives an probability inequality from hidden variable theory and then experimentally shows that it’s not valid statistically. One of the sources is titled “Bell’s theorem without inequalities”, but it doesn’t look like I can see the full paper for free, nor would I probably be able to understand it.

No one else has ventured a response, so I’ll follow up.

A standard version of the Bell’s-related experiment offers three tests (0, 120°, and 240° deflectors) on each of two entangled particles. If we pretend normal causality applies, the test results comprise six bits of pre-determined information, with three of the six bits redundant. For the GHZ experiment two tests are offered on each of three entangled photons, again for six bits total. A big difference is that three tests can be performed on the three-photon system, but only two on the two-photon system.

If we pretend that all three bits can be tested in the two-photon experiment, there are eight cases altogether: six mismatches (HHT, HTH, THH, TTH, THT, HTT, all equally likely) and two matches (HHH, TTT, equally likely). The probability of mismatch can range from 0 to 1 (implying pairwise mismatch from 0 to 2/3). In the Bell’s setup, match is intuitively impossible, so we expect each of the six mismatches to occur 1/6 of the time.

But those probabilities cannot be measured. Instead we can measure pairwise mismatch and find an “impossible” 3/4. Simple arithmetic then produces 3/16 probability for each mismatch case, and **-**1/16 for each match case. Negative probabilities are impossible; this contradiction demonstrates that the system doesn’t operate according to ordinary causality. But note that the six mismatch states of the hidden variables would all be possible; only the probabilities smell funny.

In the GHZ experiment, an H’/V’ test is performed on one photon.
If it is H’, the other photons will match each other if both are submitted to an H’/V’ test; but they will mismatch if both are submitted to an L/R test.
If it is V’, the other photons will mismatch each other if both are submitted to an H’/V’ test; but they will match if both are submitted to an L/R test.

Given these results, all 64 possible settings of the 6 “hidden” bits are impossible, no probabilistic argument is needed – all the probabilities must be zero. :smack:

This pdf paper describes an experimental setup and discussion of three-particle entanglement:

Thus we conclude that the local realistic model predicts none of the terms occurring in a quantum prediction and vice versa. This means that whenever local realism predicts that a specific result will definitely occur for a measurement on one of the photons based on the results for the other two, quantum physics definitely predicts the opposite result. For example, if two photons are both found to be H’ polarized, local realism predicts the third photon to carry polarization V’ while quantum physics predicts H’. This is the GHZ contradiction between local realism and quantum physics.

In the case of Bell’s inequalities for two photons, the conflict between local realism and quantum physics arises for statistical predictions of the theory; but for three entangled particles the conflict arises even for the definite predictions. Statistics now only results from the inevitable experimental limitations occurring in any and every experiment, even in classical physics.

A weirdness arose forming the above quote without typing from scratch. It’s irrelevant to any physics so I hide it in a Spoiler box.

Chrome browser doesn’t allow copy-paste from the pdf file so I tried a different search hoping I could find a page to copy-paste the above excerpt. This second search yielded two hits: a paywall and a webpage supposedly about poker chips?!??! That page (URL = http://noble-poker.fcpages . com/custom-poker-chips.html – I’ve inserted spaces to avoid accidental click) appears to be a malware trap which somehow grabbed from the physics paper for fill :eek: , but a copy-paste can be done from its Google cache!